Geometric permutations of disjoint unit spheres

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Geometric permutations of disjoint unit spheres

We show that a set of n disjoint unit spheres in R admits at most two distinct geometric permutations if n ≥ 9, and at most three if 3 ≤ n ≤ 8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R: if any subset of size 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.

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Hadwiger and Helly-type theorems for disjoint unit spheres

We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...

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ژورنال

عنوان ژورنال: Computational Geometry

سال: 2005

ISSN: 0925-7721

DOI: 10.1016/j.comgeo.2004.08.003